MHB How Do Vieta's Formulas Apply to Cubic Polynomial Roots?

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Vieta's formulas provide relationships between the coefficients of a cubic polynomial and its roots, allowing for the calculation of sums and products of the roots. The expressions for the sums of pairs of roots, such as $(\alpha+\beta)+(\alpha+\gamma)+(\beta+\gamma)$, can be derived using these formulas. Additionally, the product of sums of pairs of roots, represented by $(\alpha+\beta)(\alpha+\gamma)(\beta+\gamma)$, can also be evaluated through Vieta's relationships. Another expression, $(\alpha+\beta)(\alpha+\gamma)+(\beta+\gamma)(\alpha + \beta)+(\alpha+\gamma)(\beta+\gamma)$, showcases further applications of Vieta's formulas in understanding cubic polynomials. Overall, these discussions highlight the utility of Vieta's formulas in analyzing the relationships between roots of cubic polynomials.
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You can read $\alpha + \beta + \gamma $, $\alpha\beta +\alpha \gamma + \beta \gamma$ and $\alpha \beta \gamma$ off from the polynomial.

Now what's $(\alpha+\beta)+(\alpha+\gamma)+(\beta+\gamma)$?

And what's $(\alpha+\beta)(\alpha+\gamma)(\beta+\gamma)$?

And finally $(\alpha+\beta)(\alpha+\gamma)+(\beta+\gamma)(\alpha + \beta)+(\alpha+\gamma)(\beta+\gamma)$?Those questions all seem to be an exercise in Vietas formulas.
 
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